Chaos Theory

Since its inception, science relied on predictability and order. The true beauty of science was its uncanny ability to find patterns and regularity in seemingly random systems. For centuries the human mind as easily grasped and mastered the concepts of linearity. Physics illustrated the magnificent order to which the natural world obeyed. If there is a God he is indeed mathematical. Until the 19th century Physics explained the processes of the natural world successfully, for the most part. There were still many facets of the universe that were an enigma to physicists. Mathematicians could indeed illustrate patterns in nature but there were many aspects of Mother Nature that remained a mystery to Physicists and Mathematicians alike. Mathematics is an integral part of physics. It provides an order and a guide to thinking; it shows the relationship between many physical phenomenons. The error in mathematics until that point was linearity. "Clouds are not spheres, mountains are not cones, bark is not smooth, nor does lightning travel in a straight line." - Benoit Mandlebrot. Was it not beyond reason that a process, which is dictated by that regularity, could master a world that shows almost no predictability whatsoever? A new science and a new kind of mathematics were developed that could show the universe's idiosyncrasies. This new amalgam of mathematics and physics takes the order of linearity and shows how it relates to the unpredictability of the world around us. It is called Chaos Theory.

The secular definition of chaos can be misleading when the word is used in a scientific context. As defined by Webster's dictionary chaos is total disorder. That may lead one to believe that chaos theory is indeed the study of total disorder, which it truly is not. In 1986 at a prestigious conference on Chaos another definition for chaos was introduced. It is stochastic behavior occurring in a deterministic set. This definition of chaos was hesitantly brought forth. The scientists, mathematicians and intellectuals present were hesitant to define a concept they did not truly understand yet. They left the scientific community with a rather cryptic and oxymoronic definition of chaos. Deterministic sets behave by precise unbreakable law. Stochastic behavior is the opposite of deterministic it has no finite laws, it is totally dependent

upon chance. The dissected definition of chaos is lawless behavior that is ruled entirely by law. (Stewart 16-17)

The principles of Chaos Theory are complex and abstract. Perhaps the simplest and most essential ideas behind chaos theory are embodied in the aphorism known as the Butterfly Effect. The butterfly effect states that the flapping of a butterfly's wings in Hong Kong can change the weather in New York. It means that a minuscule

change in the initial conditions of a system, in this case the weather, is magnified greatly in the end conditions of that same system. The ultra sensitivity to the initial conditions of a system was not a new and striking discovery. In fact it was shown in ancient folklore;

"For want of a nail, the shoe was lost;

For want of a shoe, the horse was lost;

For want of a horse, the rider was lost;

For want of a rider, the battle was lost;

For the want of a battle, the kingdom was lost!"

The smallest variation in the initial conditions of a system can result in huge differences in concluding events. There was no nail, and because of this seemingly insignificant detail in the initial condition, the kingdom was lost. Another example of the butterfly affect is two pieces of wood floating on a river. Place those two logs at nearly the same point on the river and let them go. It is absolutely impossible to predict where those logs will be later downstream. When those logs are set on the water a slight breeze, a fish that swims underneath one of them, or even a single droplet of additional water in the initial stage can totally change the end result until no resemblance between the two is seen. (Briggs, Peat 49) There is a definite correlation between that small butterfly and a storm in New York, as well as the two logs. Chaos Theory states that within the unpredictability that makes those changes there is indeed a specific order. Chaos works in order and within all order there is chaos.

The butterfly effect as well as the two logs depends solely on iteration. Iteration is feedback that continually reabsorbs its predecessors. Iteration is a very common process, which can appear in fields as diverse as artificial intelligence or the cycling replacement of cells in the human body. (Briggs Peat 66) Iteration provides a sort of self-reference. For example the word "time" is defined with words such as "period" or "instant". Look up the definition of those words and it will eventually lead back to the word "time". (Briggs Peat 68)

MIT meteorologist Edward Lorenz has the distinction of being the first person to show how iteration creates chaos. In 1960 he was solving non linear equations on his computer that would show a model for the earth's atmosphere. He repeated a certain forecast to check his data and when he substituted the numbers in the second time he rounded off the figures to three decimal places instead of the six he received initially. He plugged in these numbers and left the computer. He returned to a surprise. The forecast before him was not a double check on his previous information, it was a totally new forecast altogether! That three decimal place difference between the two sets of numbers had been magnified greatly in the process of solving those equations. (Briggs Peat 68-69)

Just as the butterfly effect embodies the principles of Chaos Theory, a single image has become an emblem for the early pioneers of chaos. The Lorenz attractor (Figure 1) is a magical image that resembles an owl's mask or a butterfly's wings. (Gleick 29) Fig. 1

Lorenz then tried to model the chaos of a gaseous system, like the earth's atmosphere. He used his knowledge in the physics field of fluid dynamics to simplify three equations to invent the following three-dimensional system of equations:

dx/dt=delta*(y-x)

dy/dt=r*x-y-x*z

dz/dt=x*y-b*z

Where delta is an inconsequential constant for which Lorenz used a value of ten. The variable r is the difference in temperature between the top and the bottom of the gaseous system. The variable b is the width to height ratio of the box, which contains the